Theorem eulerthlem1 12227

Description: Lemma for eulerth 12233. (Contributed by Mario Carneiro, 8-May-2015.)

Hypotheses

Ref	Expression
eulerthlem1.1	|- ( ph -> ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) )
eulerthlem1.2	|- S = { y e. ( 0..^ N ) | ( y gcd N ) = 1 }
eulerthlem1.3	|- T = ( 1 ... ( phi ` N ) )
eulerthlem1.4	|- ( ph -> F : T -1-1-onto-> S )
eulerthlem1.5	|- G = ( x e. T |-> ( ( A x. ( F ` x ) ) mod N ) )

Assertion

Ref	Expression
eulerthlem1	|- ( ph -> G : T --> S )

Distinct variable groups:   x, y, A   x, F, y   x, G, y   x, N, y   x, S   ph, x, y   x, T, y

Allowed substitution hint:   S( y)

Proof of Theorem eulerthlem1

Step	Hyp	Ref	Expression
1		eulerthlem1.1	. . . . . . 7 |- ( ph -> ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) )
2	1	simp2d 1010	. . . . . 6 |- ( ph -> A e. ZZ )
3	2	adantr 276	. . . . 5 |- ( ( ph /\ x e. T ) -> A e. ZZ )
4		eulerthlem1.4	. . . . . . . . . 10 |- ( ph -> F : T -1-1-onto-> S )
5		f1of 5462	. . . . . . . . . 10 |- ( F : T -1-1-onto-> S -> F : T --> S )
6	4, 5	syl 14	. . . . . . . . 9 |- ( ph -> F : T --> S )
7	6	ffvelcdmda 5652	. . . . . . . 8 |- ( ( ph /\ x e. T ) -> ( F ` x ) e. S )
8		oveq1 5882	. . . . . . . . . 10 |- ( y = ( F ` x ) -> ( y gcd N ) = ( ( F ` x ) gcd N ) )
9	8	eqeq1d 2186	. . . . . . . . 9 |- ( y = ( F ` x ) -> ( ( y gcd N ) = 1 <-> ( ( F ` x ) gcd N ) = 1 ) )
10		eulerthlem1.2	. . . . . . . . 9 |- S = { y e. ( 0..^ N ) | ( y gcd N ) = 1 }
11	9, 10	elrab2 2897	. . . . . . . 8 |- ( ( F ` x ) e. S <-> ( ( F ` x ) e. ( 0..^ N ) /\ ( ( F ` x ) gcd N ) = 1 ) )
12	7, 11	sylib 122	. . . . . . 7 |- ( ( ph /\ x e. T ) -> ( ( F ` x ) e. ( 0..^ N ) /\ ( ( F ` x ) gcd N ) = 1 ) )
13	12	simpld 112	. . . . . 6 |- ( ( ph /\ x e. T ) -> ( F ` x ) e. ( 0..^ N ) )
14		elfzoelz 10147	. . . . . 6 |- ( ( F ` x ) e. ( 0..^ N ) -> ( F ` x ) e. ZZ )
15	13, 14	syl 14	. . . . 5 |- ( ( ph /\ x e. T ) -> ( F ` x ) e. ZZ )
16	3, 15	zmulcld 9381	. . . 4 |- ( ( ph /\ x e. T ) -> ( A x. ( F ` x ) ) e. ZZ )
17	1	simp1d 1009	. . . . 5 |- ( ph -> N e. NN )
18	17	adantr 276	. . . 4 |- ( ( ph /\ x e. T ) -> N e. NN )
19		zmodfzo 10347	. . . 4 |- ( ( ( A x. ( F ` x ) ) e. ZZ /\ N e. NN ) -> ( ( A x. ( F ` x ) ) mod N ) e. ( 0..^ N ) )
20	16, 18, 19	syl2anc 411	. . 3 |- ( ( ph /\ x e. T ) -> ( ( A x. ( F ` x ) ) mod N ) e. ( 0..^ N ) )
21		modgcd 11992	. . . . 5 |- ( ( ( A x. ( F ` x ) ) e. ZZ /\ N e. NN ) -> ( ( ( A x. ( F ` x ) ) mod N ) gcd N ) = ( ( A x. ( F ` x ) ) gcd N ) )
22	16, 18, 21	syl2anc 411	. . . 4 |- ( ( ph /\ x e. T ) -> ( ( ( A x. ( F ` x ) ) mod N ) gcd N ) = ( ( A x. ( F ` x ) ) gcd N ) )
23	17	nnzd 9374	. . . . . 6 |- ( ph -> N e. ZZ )
24	23	adantr 276	. . . . 5 |- ( ( ph /\ x e. T ) -> N e. ZZ )
25	16, 24	gcdcomd 11975	. . . 4 |- ( ( ph /\ x e. T ) -> ( ( A x. ( F ` x ) ) gcd N ) = ( N gcd ( A x. ( F ` x ) ) ) )
26	23, 2	gcdcomd 11975	. . . . . . 7 |- ( ph -> ( N gcd A ) = ( A gcd N ) )
27	1	simp3d 1011	. . . . . . 7 |- ( ph -> ( A gcd N ) = 1 )
28	26, 27	eqtrd 2210	. . . . . 6 |- ( ph -> ( N gcd A ) = 1 )
29	28	adantr 276	. . . . 5 |- ( ( ph /\ x e. T ) -> ( N gcd A ) = 1 )
30	24, 15	gcdcomd 11975	. . . . . 6 |- ( ( ph /\ x e. T ) -> ( N gcd ( F ` x ) ) = ( ( F ` x ) gcd N ) )
31	12	simprd 114	. . . . . 6 |- ( ( ph /\ x e. T ) -> ( ( F ` x ) gcd N ) = 1 )
32	30, 31	eqtrd 2210	. . . . 5 |- ( ( ph /\ x e. T ) -> ( N gcd ( F ` x ) ) = 1 )
33		rpmul 12098	. . . . . 6 |- ( ( N e. ZZ /\ A e. ZZ /\ ( F ` x ) e. ZZ ) -> ( ( ( N gcd A ) = 1 /\ ( N gcd ( F ` x ) ) = 1 ) -> ( N gcd ( A x. ( F ` x ) ) ) = 1 ) )
34	24, 3, 15, 33	syl3anc 1238	. . . . 5 |- ( ( ph /\ x e. T ) -> ( ( ( N gcd A ) = 1 /\ ( N gcd ( F ` x ) ) = 1 ) -> ( N gcd ( A x. ( F ` x ) ) ) = 1 ) )
35	29, 32, 34	mp2and 433	. . . 4 |- ( ( ph /\ x e. T ) -> ( N gcd ( A x. ( F ` x ) ) ) = 1 )
36	22, 25, 35	3eqtrd 2214	. . 3 |- ( ( ph /\ x e. T ) -> ( ( ( A x. ( F ` x ) ) mod N ) gcd N ) = 1 )
37		oveq1 5882	. . . . 5 |- ( y = ( ( A x. ( F ` x ) ) mod N ) -> ( y gcd N ) = ( ( ( A x. ( F ` x ) ) mod N ) gcd N ) )
38	37	eqeq1d 2186	. . . 4 |- ( y = ( ( A x. ( F ` x ) ) mod N ) -> ( ( y gcd N ) = 1 <-> ( ( ( A x. ( F ` x ) ) mod N ) gcd N ) = 1 ) )
39	38, 10	elrab2 2897	. . 3 |- ( ( ( A x. ( F ` x ) ) mod N ) e. S <-> ( ( ( A x. ( F ` x ) ) mod N ) e. ( 0..^ N ) /\ ( ( ( A x. ( F ` x ) ) mod N ) gcd N ) = 1 ) )
40	20, 36, 39	sylanbrc 417	. 2 |- ( ( ph /\ x e. T ) -> ( ( A x. ( F ` x ) ) mod N ) e. S )
41		eulerthlem1.5	. 2 |- G = ( x e. T |-> ( ( A x. ( F ` x ) ) mod N ) )
42	40, 41	fmptd 5671	1 |- ( ph -> G : T --> S )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 978   = wceq 1353   e. wcel 2148   {crab 2459   |-> cmpt 4065   -->wf 5213   -1-1-onto->wf1o 5216   ` cfv 5217  (class class class)co 5875   0cc0 7811   1c1 7812   x. cmul 7816   NNcn 8919   ZZcz 9253   ...cfz 10008  ..^cfzo 10142   mod cmo 10322   gcd cgcd 11943   phicphi 12209

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-mulrcl 7910  ax-addcom 7911  ax-mulcom 7912  ax-addass 7913  ax-mulass 7914  ax-distr 7915  ax-i2m1 7916  ax-0lt1 7917  ax-1rid 7918  ax-0id 7919  ax-rnegex 7920  ax-precex 7921  ax-cnre 7922  ax-pre-ltirr 7923  ax-pre-ltwlin 7924  ax-pre-lttrn 7925  ax-pre-apti 7926  ax-pre-ltadd 7927  ax-pre-mulgt0 7928  ax-pre-mulext 7929  ax-arch 7930  ax-caucvg 7931

This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-if 3536  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-po 4297  df-iso 4298  df-iord 4367  df-on 4369  df-ilim 4370  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-frec 6392  df-sup 6983  df-pnf 7994  df-mnf 7995  df-xr 7996  df-ltxr 7997  df-le 7998  df-sub 8130  df-neg 8131  df-reap 8532  df-ap 8539  df-div 8630  df-inn 8920  df-2 8978  df-3 8979  df-4 8980  df-n0 9177  df-z 9254  df-uz 9529  df-q 9620  df-rp 9654  df-fz 10009  df-fzo 10143  df-fl 10270  df-mod 10323  df-seqfrec 10446  df-exp 10520  df-cj 10851  df-re 10852  df-im 10853  df-rsqrt 11007  df-abs 11008  df-dvds 11795  df-gcd 11944

This theorem is referenced by:  eulerthlemh  12231  eulerthlemth  12232